As has been indicated, bus drivers don't (officially) accept cash within central London (although some drivers will occasionally do so). You're meant to buy a ticket from the machines at bus stops (unless you use an Oyster card or Travelcard).
If you buy tickets from the machines you'll pay £2.40 per journey, irrespective of distance. If you use an Oyster card, £1.40 will be deducted from the credit on it for each journey (but only up to a maximum of £4.40 per day).
If you've got a 'contactless' debit card you can also use it to pay on the bus. You'll only pay the same fare as with an Oyster card (i.e. £1.40) but there's no daily cap on the amount you can pay.
So, for example, 10 bus journeys in a single day would cost you £24.00 if you bought tickets from the machines, £14.00 if you used contactless payment or £4.40 using an Oyster card.
Oyster cards are available from all Underground stations, plus some National Rail stations, as well as from TfL information offices and other outlets. There's a £5 deposit to get hold of one and you'd need to put some credit on it as well. However you can easily get your deposit back (together with any unused credit) by handing in your card at any Oyster card sales outlet:
http://www.tfl.gov.uk/assets/downloads/tickets/getting-around-with-oyster.pdf
Lastly, if you're travelling into London by rail, just for a day, it's probably easiest to buy a rail ticket which includes a Travelcard, giving you unlimited travel on London bus and underground services.